Flavor Changing Effects in Theories with a Heavy Z' Boson with Family Non-Universal Couplings

20 Pages·0.24 MB·English

by  Paul Langacker

UPR-0870-T
hep-ph/0001204
Flavor Changing Effects in Theories with a Heavy Z Boson
′
with Family Non-Universal Couplings
0 Paul Langacker and Michael Plu¨macher
0
0 Department of Physics and Astronomy
2
University of Pennsylvania, Philadelphia PA 19104-6396, USA

n
a
J

1
Abstract
3

2 There are theoretical and phenomenological motivations that there may exist addi-
v tional heavy Z′ bosons with family non-universal couplings. Flavor mixing in the quark
4 and lepton sectors will then lead to flavor changing couplings of the heavy Z′, and also of
0
2 the ordinary Z when Z Z′ mixing is included. The general formalism of such effects is
−
1 described, andapplications are made to a variety of flavor changing and CP-violating tree
0
and loop processes. Results are described for three specific cases motivated by a specific
0
0 heterotic string model and by phenomenological considerations, including cases in which
/ all three families have different couplings, and those in which the first two families, but
h
p notthethird,havethesamecouplings. Evenwithinaspecifictheorytheresultsaremodel
-
dependent because of unknown quark and lepton mixing matrices. However, assuming
p
e that typical mixings are comparable to the CKM matrix, processes such as coherent µ e
h conversion in a muonic atom, K0 K0 and B B mixing, ǫ, and ǫ′/ǫ lead to significa−nt
:
v constraints on Z′ bosons in the t−heoretically −and phenomenologically motivated range
i
X MZ′ 1 TeV.
∼
r
a
1 Introduction
Additional heavy neutral Z′ gauge bosons are amongst the best motivated extensions of the
standard model (SM), or its supersymmetric extension (MSSM) [1]. In particular, they often
occur in grand unified theories (GUTs), superstring theories, and theories with large extra
dimensions [2]. In traditional GUTs, the scale of the Z′ mass is arbitrary. However, in pertur-
bative heterotic string models with supergravity mediated supersymmetry breaking, the U(1)′
and electroweak breaking are both driven by a radiative mechanism, with their scales set by
the soft supersymmetry breaking parameters, implying that the Z′ mass should be less than
around a TeV [3]. (The breaking can be at a larger intermediate scale if it is associated with
a D-flat direction [4].) Furthermore, the extra symmetry can forbid an elementary µ term,
while allowing an effective µ and Bµ to be generated at the U(1)′ breaking scale, providing a
solution to the µ problem without introducing cosmological problems [5, 3]. An extra U(1)′
provides an analogous solution to the µ problem in models of gauge-mediated supersymmetry
breaking [6, 7]. An extra U(1)′ gauge symmetry does not by itself spoil the successes of gauge
coupling unification.
There are stringent limits on the mass of an extra Z′ from the non-observation of direct
production followed by decays into e+e− or µ+µ− by CDF [8], while indirect constraints from
precision data also limit the Z′ mass (weak neutral current processes and LEP II) and severely
constrain the Z Z′ mixing angle θ (Z-pole) [9]. These limits are model-dependent, but are
−
typically in the range MZ′ > O(500) GeV and θ < few 10−3 for standard GUT models.
| | ×
Recently, it has been argued [10] that both Z-pole data and atomic parity violation are much
better described if the SM or the MSSM is extended by an additional heavy Z′.
There is thus both theoretical and experimental motivation for an additional Z′, most
likely in the range 500 GeV - 1 TeV. If true, there should be a good chance to observe it at
RUN II at the Tevatron, and certainly at the LHC. Also, in this mass range, it should be
possible to carry out significant diagnostic probes of the Z′ couplings at the LHC and at a
future NLC [12], which would complement those from the precision experiments [10]. The
existence of a heavy Z′ would also suggest a spectrum of sparticles considerably different than
most versions of the MSSM [13].
Most studies have assumed that the Z′ gauge couplings are family universal [14], so that
they remain diagonal even in the presence of fermion flavor mixing by the GIM mechanism.
However, in string models it is possible to have family-nonuniversal Z′ couplings, because of
different constructions of the different families. For instance, the consequences of a model by
Chaudhuri, Hockney, and Lykken [15] have been extensively analyzed in [16], where it was
shown that most F and D-flat directions involve an additional U(1)′ broken at the TeV scale.
Inthiscase, thethirdgenerationquarkcouplingsaredifferent fromthefirst twofamilies, andall
three lepton generations have different couplings. Other aspects of the model are not realistic,
but nevertheless this provides a motivation to consider non-universal couplings. Similarly,
¯
possible anomalies in the Z-pole bb asymmetries [17] suggest that the data are better fitted
with a non-universal Z′ [10].
Family-nonuniversal Z′ couplings necessarily lead to flavor-changing (non-diagonal) Z′
couplings, and possibly to new CP-violating effects, when quark and lepton flavor mixing are
2
taken into account1. This will also imply flavor violating Z couplings if there is Z Z′ mixing.
−
Thus, flavor changing neutral current (FCNC) and CP-violating effects should be considered an
additional constraint, consequence, or possibly diagnostic probe of an extra Z′, and conversely
as another motivation to search for FCNC effects.
Even for a model in which the Z′ couplings are specified (prior to flavor mixing), the
predictions for FCNC are still model dependent, because they depend on the individual unitary
transformations for the left (L) and right (R) chiral u-quarks, d-quarks, charged leptons, and
neutrinos which diagonalize their respective mass matrices. However, only thecombination ofL
matrices for the u and d occurring in the Cabibbo-Kobayashi-Maskawa (CKM) matrix is known
experimentally (with weak constraints on the leptonic analog fromneutrino oscillations), so one
cannot make definitive predictions. We will present our results for arbitrary mixings, and then
illustrate assuming that all of the mixing matrices are comparable to the CKM matrix2.
Inthenextsectionwediscussthegeneralformalismandintroduceournotation. Insection
3 we discuss Z′ contributions to flavor changing processes forbidden in the standard model and
to new contributions to SM processes, and we use experimental results to constrain the Z′
couplings. Specific examples of a Z′ with flavor changing couplings are discussed in section 4.
Models in which all three families have different Z′ charges in the gauge eigenstate basis are
strongly constrained by experimental results, and even models in which the first two families
have the same couplings, but not the third, can yield flavor changing rates above experimental
limits unless they are suppressed by small mixing elements. Finally, in section 5 we summarize
our results and present our conclusions.
2 Formalism
We will use the formalism developed in ref. [19] and generalize it to the case of flavor violating
Z′ couplings. In the basis in which all the fields are gauge eigenstates the neutral current
Lagrangian is given by
= eJµ A g J(1)µZ0 g J(2)µZ0 , (1)
LNC − em µ − 1 1,µ − 2 2,µ
where Z0 is the SU(2) U(1) neutral gauge boson, Z0 the new gauge boson associated with
1 × 2
an additional Abelian gauge symmetry, and the currents are
J(1) = ψ γ [ǫ (i)P +ǫ (i)P ]ψ , (2)
µ i µ L L R R i
i
X
J(2) = ψ γ ǫ(2) P +ǫ(2) P ψ , (3)
µ i µ ψLij L ψRij R j
Xi,j h i
1 ′
ModelswithanextraZ oftenincludeadditionalexoticfermions(i.e.,withnon-standardSU(2)assignments)
′
as well. Mixing of ordinary and exotic fermions could lead to flavor changing Z and Z couplings even in the
absenceoffamilynon-universalcharges,andtonon-universalW couplings[18]. Wedonotconsidersucheffects
in this paper.
2ForacompletetheorytheU(1)′chargeswouldconstrainthepossibleflavormixings. However,suchrelations
would be much more specific than the general issues considered here.
3
(2)
where the sum extends over all quarks and leptons ψ and P = (1 γ )/2. ǫ denote
i,j R,L ± 5 ψR,Lij
the chiral couplings of the new gauge boson, and the standard model chiral couplings are
ǫ (i) = sin2θ Q , ǫ (i) = ti sin2θ Q , (4)
R − W i L 3 − W i
where ti and Q are the third component of the weak isospin and the electric charge of fermion
3 i
i, respectively. g = g/cosθ = e/sinθ and g are the gauge couplings of the two U(1)
1 W W 2
factors.
Flavor changing effects (FCNCs) immediately arise if the ǫ(2) are non-diagonal matrices.
If the Z0 couplings are diagonal but non-universal, flavor changing couplings are induced by
2
fermion mixing. The fermion Yukawa matrices h in the weak eigenstate basis can be diago-
ψ
nalized by unitary matrices Vψ
R,L
h = Vψh Vψ† , (5)
ψ,diag R ψ L
where the CKM matrix is given by the combination
V = VuVd† . (6)
CKM L L
Hence, the chiral Z0 couplings in the fermion mass eigenstate basis read
2
BψL Vψǫ(2)Vψ† , and BψR Vψǫ(2)Vψ† . (7)
ij ≡ L ψL L ij ≡ R ψR R
(cid:18) (cid:19)ij (cid:18) (cid:19)ij
Further, Z Z′ mixing is induced by electroweak symmetry breaking, implying that Z0
− 1,2
are related to mass eigenstates by an orthogonal transformation. Hence, the couplings of the
massive gauge boson mass eigenstates Z(i) are
g g
Z = g cosθJ(1)µ + 2 sinθJ(2)µ Z(1) g 2 cosθJ(2)µ sinθJ(1)µ Z(2) , (8)
LNC − 1" g1 # µ − 1"g1 − # µ
where θ is the Z Z′ mixing angle. The standard model weak neutral current J(1)µ is given
in eq. (2), and J(2−)µ has a form analogous to eq. (3), with the ǫ(2) replaced by the couplings
ψR,L
BψR,L from eq. (7).
We have neglected kinetic mixing [20], since it only amounts to a redefinition of the
unknown Z′ couplings.3
At low energies, the effective four-fermion interactions are then given by
4G
= F ρ J(1)2 +2wJ(1) J(2) +yJ(2)2 (9)
eff eff
−L √2 ·
Xψ,χ(cid:16) (cid:17)
4G
= F CijQij +CijQij +DijOij +DijOij , (10)
√2 kl kl kl kl kl kl kl kl
Xψ,χi,Xj,k,lh i
3Kinetic mixing allows the redefined Z0 charges to haveea ceomponent of weafk hyepercharge, which would
2
otherwise not be allowed. This is irrelevant for the purposes of this paper.
4
with the local operators4
Qij = ψ γµP ψ (χ γ P χ ) , Qij = ψ γµP ψ (χ γ P χ ) , (11)
kl i L j k µ L l kl i R j k µ R l
(cid:16) (cid:17) (cid:16) (cid:17)
Oij = ψ γµP ψ (χ γ P χ ) , Oeij = ψ γµP ψ (χ γ P χ ) .
kl i L j k µ R l kl i R j k µ L l
(cid:16) (cid:17) (cid:16) (cid:17)
ψ and χ represent classes of fermions with the sameeSM quantum numbers, i.e., u, d, e−, and
ν, while i,j,k,l are family indices. The coefficients are
Cij = ρ δ δ ǫ (i)ǫ (k)+wδ ǫ (ψ )BχL +wδ ǫ (χ )BψL +yBψLBχL , (12)
kl eff ij kl L L ij L i kl kl L l ij ij kl
Cij = ρ δ δ ǫ (i)ǫ (k)+wδ ǫ (ψ )BχR +wδ ǫ (χ )BψR +yBψRBχR , (13)
kl eff ij kl R R ij R i kl kl R l ij ij kl
Deij = ρ δ δ ǫ (i)ǫ (k)+wδ ǫ (ψ )BχR +wδ ǫ (χ )BψL +yBψLBχR , (14)
kl eff ij kl L R ij L i kl kl R l ij ij kl
Dij = ρ δ δ ǫ (i)ǫ (k)+wδ ǫ (ψ )BχL +wδ ǫ (χ )BψR +yBψRBχL . (15)
kl eff ij kl R L ij R i kl kl L l ij ij kl
The coeffifcients are given by
M2
ρ = ρ cos2θ +ρ sin2θ , ρ = W , (16)
eff 1 2 i M2cos2θ
i W
g
2
w = sinθcosθ(ρ ρ ) , (17)
1 2
g −
1
2
g
y = 2 (ρ sin2θ +ρ cos2θ) , (18)
1 2
g1!
where M are the masses of the neutral gauge boson mass eigenstates and θ is the electroweak
i W
mixing angle.
3 Flavor Changing Processes
In this section we will discuss flavor violating processes forbidden in the SM and new contri-
butions to SM processes. Experimental bounds or results on these processes (cf. ref. [17]) can
then be used to constrain the Z′ couplings.
3.1 Z Decays
Due to the Z Z′ mixing, Z couples to J(2). The decay width for a flavor changing Z decay
−
at tree level is given by
Γ(Z ψ ψ¯ ) = CGFρ1M13 g2 2sin2θ BψL 2 + BψR 2 , (19)
→ i j 3√2π g1! (cid:18)(cid:12) ij (cid:12) (cid:12) ij (cid:12) (cid:19)
4These operators are not all independent. For couplings of four fe(cid:12)(cid:12)rmion(cid:12)(cid:12)s of t(cid:12)(cid:12)he sa(cid:12)(cid:12)me type, ψ =χ, e.g. four
charged leptons, one has Qij =Qkl, Qij =Qkl and Oij =Okl.
kl ij kl ij kl ij
e e e
5
where C = 1 (C = 3) is the color factor for leptons (quarks). Due to strong experimental
constraintsontheZ Z′ mixingangleθ, cf.ref.[9,10], theBψR,L cannotbestronglyconstrained
−
from flavor violating Z decays.
3.2 Lepton Decays
In the SM each lepton generation has a separately conserved lepton number, if one neglects
small effects from non-vanishing neutrino masses and non-perturbative effects. The effective
Lagrangian(10)givesrisetoleptonfamilynumber violatingprocesses, althoughthetotallepton
number is still conserved.
Consider first the decay of a charged lepton l into three different charged leptons l , l
j i k
and l . At tree level, the decay width is
l
G2m5
Γ(l l l l ) = F lj Clilj +Clklj 2 + Clilj +Clklj 2+
j → i k l 48π3 lkll lill lkll lill
(cid:18)(cid:12) (cid:12) (cid:12) (cid:12)
+ Dlilj 2(cid:12)(cid:12)+ Dlklj 2 +(cid:12)(cid:12)Dli(cid:12)(cid:12)lje2 + Dleklj 2(cid:12)(cid:12) , (20)
lkll lill lkll lill
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)f (cid:12) (cid:12)f (cid:12)
where we have neglected the masses o(cid:12)f the(cid:12)final(cid:12)state(cid:12)s lep(cid:12)tons.(cid:12)If tw(cid:12)o lep(cid:12)tons in the final state
are equal (i = k), taking permutations of the external fermion lines into account yields [21]
G2m5
Γ(l l l l ) = F lj 2 Clilj 2 +2 Clilj 2 + Dlilj 2 + Dlilj 2 . (21)
j → i i l 48π3 lill lill lill lill
(cid:18) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12) (cid:12)e (cid:12) (cid:12) (cid:12) (cid:12)f (cid:12)
Since such processes are free of hadron(cid:12)ic un(cid:12)certai(cid:12)nties(cid:12)and(cid:12)well(cid:12)cons(cid:12)train(cid:12)ed experimentally
[17, 21, 22], they yield strong constraints on the leptonic couplings of a Z′ 5.
The strongest constraint on the Z′-µ-e coupling, however, comes from coherent µ-e con-
version in a muonic atom [24]. The branching fraction for this process, i.e., the ratio of the
coherent µ-econversion ratetotheµcapturerateforanucleus ofatomicnumberZ andneutron
number N is given by [25, 21]
B(µ−N e−N) = G2Fα3m5µ Ze4ff F 2 BlL 2 + BlR 2 w 1(Z N) 2Zsin2θ +
→ 2π2ΓCAPT Z | P| (cid:18)(cid:12) 12(cid:12) (cid:12) 12(cid:12) (cid:19)(cid:12)(cid:12) (cid:20)2 − − W(cid:21)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 2
(cid:12)
+y (2Z +N)(BuL +BuR)+(Z +2N) BdL +BdR , (22)
11 11 11 11
(cid:12)
h (cid:16) (cid:17)i(cid:12)
(cid:12)
(cid:12)
where Γ is the µ capture rate, Z an effective atomic charge obtained b(cid:12)y averaging the
CAPT eff
muon wave function over the nucleon, and F is a nuclear matrix element.
P
5In ref.[23] these processeswere consideredin the case of vanishing Z-Z′ mixing, Z′ couplings of the V A
′ −
form, and assuming that the Z has no diagonal couplings.
6
3.3 Radiative Decays
Neutral current penguins give rise to radiative lepton decays. Neglecting the mass of the final
state lepton, the decay width is
αG2m5
Γ(l l γ) = F lj ξlilj 2 + ξlilj 2 , (23)
j → i 8π4 L R
(cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
where the dipole moment couplings of an on-shell p(cid:12) hot(cid:12)on to(cid:12) the(cid:12)chiral µ-e currents are
1 y
ξlilj = m Dlklj = BlRm BlL +wǫ (l )BlR , (24)
L m lk lilk m l ij L j ij
lj Xk lj (cid:16) (cid:17)
1 y
ξlilj = m Dlklj = BlLm BlR +wǫ (l )BlL , (25)
R m lk lilk m l ij R j ij
lj Xk lj (cid:16) (cid:17)
f
where m is the charged lepton mass matrix.
l
A similar result holds for the decay b sγ. Since the b-quark mass is much larger than
→
the QCD-scale Λ, long-range strong interaction effects are not expected to be important in the
inclusive decay B X γ [26]. Hence, the rate for this process is usually approximated by
s
→
considering the ratio
Γ(B X γ) Γ(b sγ)
s
R → → . (26)
≡ Γ(B X eν ) ≈ Γ(b ceν )
c e e
→ →
Neglecting SM contributions, the contribution to R from the one-loop neutral current penguin
diagrams is
R = 8α V −2 f−1 m2c ξsb 2 + ξsb 2 , (27)
3π | cb| m2b! (cid:18)(cid:12) L (cid:12) (cid:12) R (cid:12) (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
where f is the phase-space factor in the semi-leptonic(cid:12) b-d(cid:12)ecay(cid:12): (cid:12)
f(x) = 1 8x+8x3 x4 12x2lnx . (28)
− − −
In analogy to eqs. (24) and (25) the flavor violating effective couplings ξsb are given by:
R,L
1 y
ξsb = m Ddkb = BdRm BdL +wǫ (b)BdR , (29)
L m dk sdk m d 23 L 23
b Xk b (cid:16) (cid:17)
1 y
ξsb = m Ddkb = BdLm BdR +wǫ (b)BdL , (30)
R m dk sdk m d 23 R 23
b Xk b (cid:16) (cid:17)
f
where d stands for the k-th generation down-type quark and m is the diagonal mass matrix
k d
of down quarks.
3.4 Leptonic Meson Decays
Meson decays can be used to place limits on the Z′ couplings to quarks. Consider the lepton
family number violating decay of a neutral pseudoscalar meson P0 into two charged leptons
7
P0 βlilj βlilj
L R
π0 Dlilj Clilj Dlilj +Clilj Clilj Dlilj Clilj +Dlilj
uu − uu − dd dd uu − uu − dd dd
K0 Dlilj +Dlilj Clilj Clilj eDlilj fDlilj +eClilj +fClilj
L ds sd − ds − sd − ds − sd ds sd
D0 √2 Dlilj Clilj f √2 fDlilj +eClilj e
cu cu cu cu
− −
(cid:16) (cid:17) (cid:16) (cid:17)
B0 √2 Dlilj Clilj √2 Dflilj +Celilj
bd − bd − bd bd
(cid:16) (cid:17) (cid:16) (cid:17)
B0 √2 Dlilj Clilj √2 Dflilj +Celilj
s bs − bs − bs bs
(cid:16) (cid:17) (cid:16) (cid:17)
f e
Table 1: Coefficients in the decay widths of pseudoscalar mesons P0. Since K0 is a linear
L
combination of K0 and K0, the β for K0 decays depend only on the real part of the Z′-d-s
R,L L
axial vector coupling Re(BdR BdL).
12 − 12
l and l , with i = j. Due to the hierarchy of lepton masses, we can neglect the mass of the
i j
6
lighter lepton. Assuming that m m (the case m m can be obtained by exchanging
lj ≪ li li ≪ lj
the lepton indices i and j in the following) the decay width is
Γ(P0 l l ) = 2Γ(P− → liνi) βlilj 2 + βlilj 2 , (31)
→ i j V 2 L R
| kl| (cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
where we have used isospin symmetry to relate the amplitude for P0 l l to the amplitude
i j
→
for the SM decay P− l ν , and V is the element of the CKM-matrix appearing in this SM
i i kl
→
process. The coefficients β for the decays we have considered are given in table 1.
R,L
In the SM the decay of a pseudoscalar P0 into a lepton and its anti-lepton is suppressed
by the GIMmechanism andcan only occur at one-looplevel (cf., e.g., ref. [27] fora discussion of
K0 l l in the SM), whereas the Z′ couplings allow tree-level contributions to such processes.
L → i i
Neglecting SM contributions, which are formally of higher order in the couplings, the decay
width reads
Γ(P0 l l ) = 4Γ(P− → liνi) m3P m2P −4m2li βlili βlili 2 , (32)
→ i i V 2 (mq2 m2)2 L − R
| kl| P − li (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where the mass dependent factor corrects for the different phase spaces in the decays of P0 and
P−, and the couplings β can again be found in table 1.
R,L
Similar formulae hold for semi-leptonic τ decays. For the process τ l π0 we have
i
→
Γ(τ l π0) = 2Γ(τ → ντπ−) βliτ 2 + βliτ 2 , (33)
→ i V 2 L R
| ud| (cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
where we have neglected the mass of the final state lepton and the β are given in the first
R,L
8
P δlilj δlilj
L R
K0 Dlilj Dlilj +Clilj Clilj Dlilj Dlilj +Clilj Clilj
L ds − sd ds − sd sd − ds sd − ds
D0 √2 Dlilj +Clilj f √2fDlilj +eClilj e
cu cu cu cu
(cid:16) (cid:17) (cid:16) (cid:17)
B0 √2 Dlilj +Clilj √2 Dflilj +Celilj
bs bs bs bs
(cid:16) (cid:17) (cid:16) (cid:17)
f e
Table 2: Coefficients for semi-leptonic meson decays. The δ for K0 decays are proportional
R,L L
to the imaginary part of the Z′-d-s vector coupling Im(BdR +BdL).
12 12
line of table 1, whereas for τ l K0 one finds
i
→
Γ(τ l K0) = 4Γ(τ → ντK−) Cliτ Dliτ 2 + Cliτ Dliτ 2 . (34)
→ i V 2 ds − ds ds − ds
| us| (cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12) (cid:12)e f (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
0
Replacing the indices d and s yields the decay width for τ l K .
i
→
3.5 Semi-Leptonic Meson Decays
All the processes discussed in the last section constrain only couplings of the form Dlilj
qkql −
Clilj , i.e., the axial vector couplings in the quark current. Limits on the corresponding vector
qkql
couplings can be obtained by considering decays of P0 into another pseudoscalar meson and
two leptons.
Particularly interesting are lepton flavor conserving, CP violating contributions to decays
K0 π0¯ll, since the branching ratios are expected to be small in the SM [27, 28] and new
L →
limits from KTeV [29] allow the imaginary part of the Z′-d-s vector coupling to be constrained.
Neglecting the electron mass but taking the µ-mass into account, the decay widths for the
semi-leptonic K0 decays considered are
L
Γ(K+ e+ν π0)
Γ(K0 e+e−π0) = 2 → e δee 2 + δee 2 , (35)
L → V 2 | L | | R |
us
| | (cid:16) (cid:17)
Γ(K+ µ+ν π0)
Γ(K0 µ+µ−π0) = 2 → µ 0.57 δµµ 2 + δµµ 2 0.48Re(δµµδµµ∗) (,36)
L → V 2 | L | | R | − L R
us
| | h (cid:16) (cid:17) i
where the numerical coefficients in the last decay width arise due to the different phase spaces
for the processes K0 µ+µ−π0 and K+ µ+ν π0, and the couplings δ for these processes
L → → µ R,L
are given in table 2. We also considered the lepton flavor violating decay
Γ(K+ µ+ν π0)
Γ(K0 µ+e−π0) = 2 → µ δeµ 2 + δeµ 2 , (37)
L → V 2 | L | | R |
us
| | (cid:16) (cid:17)
9
although experimental bounds on the Z′-µ-e coupling from coherent µ-e conversion imply that
the branching ratio for this process is several orders of magnitude below the experimental
bounds.
Similarly, for semi-leptonic D0 and B0 decays one has
Γ(D+ l ν π0)
Γ(D0 l l π0) = → j j δlilj 2 + δlilj 2 , (38)
→ i j V 2 | L | | R |
cd
| | (cid:16) (cid:17)
Γ(B+ l ν D0) f(m2 /m2 )
Γ(B0 l l K0) = → i i K B δlilj 2 + δlilj 2 , (39)
→ i j V 2 f(m2 /m2 ) | L | | R |
| cb| D B (cid:16) (cid:17)
where the phase-space function f is given in eq. (28).
3.6 Mass Splittings and CP Violation
The effective Lagrangian (10) also contributes to the mass splitting in a neutral pseudo-scalar
meson system. Again denoting the flavor eigenstates of a meson by P0 and P 0, the mass
splitting ∆m is given by
P
∆m = 2Re P0 P0 . (40)
P eff
− h |L | i
The relevant hadronic matrix elements of the operators (11) have been determined in the
vacuum insertion approximation using PCAC [30]. Hence, for a meson with the quark content
P0 = q q we obtain the following contribution to the mass splitting:
j i
2
∆m = 4√2G m F2y 1Re BqL 2 + BqR 2 1 + 1 mP Re BqLBqR ,
P F P P 3 (cid:20)(cid:16) ij (cid:17) (cid:16) ij (cid:17) (cid:21)−2 3 mqi +mqj!  (cid:16) ij ij (cid:17)
  (41)
 
where m and F are the mass and decay constant of the meson, respectively.
P P
Further, phases intheZ′ couplingsBψR,L willcontributetoCPviolatingprocesses. Limits
ij
ontheimaginarypartsofthes-d-Z′couplingscanbeplacedbyconsideringindirectCPviolation
ε in the neutral kaon system:
K
1 Im K0 K0
eff
ε = h |L | i (42)
| K| 2√2Re K0 K0
eff
h |L | i
= 2GFmKFK2y 1Im BdL 2 + BdR 2 1 + 1 mP 2 Im BdLBdR (.43)
∆mK (cid:12)(cid:12)3 (cid:20)(cid:16) 12(cid:17) (cid:16) 12 (cid:17) (cid:21)−"2 3 (cid:18)md +ms(cid:19) # (cid:16) 12 12 (cid:17)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Direct CP violation ε′ in th(cid:12)e decays K ππ can be expressed in terms of the decay ampli(cid:12)tudes
→
A = A(K (ππ) ) and A = A(K (ππ) ), where the indices 0 and 2 denote the isospin of
0 0 2 2
→ →
the final two pion state (see ref. [26] for a review):
1 ω 1
ε′ = ImA ImA eiφ˜ , (44)
−√2ReA 0 − ω 2
0 (cid:18) (cid:19)
10